Spin-spin correlators on the β/β⋆ boundaries in 2D Ising-like models: Exact analysis through theory of block Toeplitz determinants

Abstract

In this work, we investigate quantitative properties of correlation functions on the boundaries between two 2D Ising-like models with dual parameters β and β⋆. Spin-spin correlators in such constructions without reflection symmetry with respect to translation-invariant directions are usually represented as 2×2 block Toeplitz determinants which are usually significantly harder than the scalar (1×1 block) versions. Nevertheless, we show that for the specific β/β⋆ boundaries considered in this work, the symbol matrices allow explicit commutative Wiener-Hopf factorizations. As a result, the constants E(a) and E(a˜) for the large n asymptotics still allow explicit representations that generalize the strong Szegö's theorem for scalar symbols. However, the Wiener-Hopf factors at different z do not commute. We will show that due to this non-commutativity, “logarithmic divergences” in the Wiener-Hopf factors generate certain “anomalous terms” in the exponential form factor expansions of the re-scaled correlators. Since our boundaries in the naive scaling limits can be formulated as certain integrable boundaries/defects in 2D massive QFTs, the results of this work facilitate detailed comparisons with bootstrap approaches.